Solving Orthogonal Matrix Homework w/ Symmetric Matrix

[beetle2]

Homework Statement

Given the symmetric Matrix

1 2
2 5

find an orthogonal matrix P such that C=BAB^t

Homework Equations

The Attempt at a Solution

I found the eigenvalues to be $3-(2\sqrt{2})$ and $3+(2\sqrt{2})$

giving eigenvectors of
$[1,1-\sqrt{2}]$ and $[1,1+\sqrt{2}]$

As the dot product of these vectors is 0 they are orthogonal.

do I just normalise each vector and use them as the column vectors of P?

 

[aPhilosopher]

It's going to be very difficult to make a statement about C=BAB^t in general. Knowing a symmetric matrix P, associated with an unlabeled matrix does very little to help.

 

[beetle2]

Sorry the matrix is

A =

1 2
2 5

find an orthogonal matrix P such that C=PAP^t where C is diagonal

regards

 

[HallsofIvy]

Then find the eigenvectors of A. P will be an orthogonal matrix with the eigenvectors of A as rows.

 

[beetle2]

So I multiply P =
$\[ \left( \begin{array}{cc} 1 & 1-\sqrt{2} \\ 1 & 1+\sqrt{2} \\ \end{array} \right)\]$
by A =
$\[ \left( \begin{array}{cc} 1 & 2 \\ 2 & 5 \\ \end{array} \right)\]$
and P^T =
$\[ \left( \begin{array}{cc} 1 & 1 \\ 1-\sqrt{2} & 1+\sqrt{2} \\ \end{array} \right)\]$

which gives C =

$\[ \left( \begin{array}{cc} 20-14\sqrt{2} & 0 \\ 0 & 14\sqrt{2}+20 \\ \end{array} \right)\]$


Is this right? I know that C is diagonal but isn't it supposed to have the eigenvalues on the main diagonal?
regards